| L(s) = 1 | − 2-s + 0.280·3-s + 4-s + 0.751·5-s − 0.280·6-s + 1.30·7-s − 8-s − 2.92·9-s − 0.751·10-s − 1.75·11-s + 0.280·12-s − 2.70·13-s − 1.30·14-s + 0.210·15-s + 16-s − 4.51·17-s + 2.92·18-s + 5.21·19-s + 0.751·20-s + 0.367·21-s + 1.75·22-s + 7.11·23-s − 0.280·24-s − 4.43·25-s + 2.70·26-s − 1.65·27-s + 1.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.161·3-s + 0.5·4-s + 0.336·5-s − 0.114·6-s + 0.495·7-s − 0.353·8-s − 0.973·9-s − 0.237·10-s − 0.528·11-s + 0.0809·12-s − 0.748·13-s − 0.350·14-s + 0.0543·15-s + 0.250·16-s − 1.09·17-s + 0.688·18-s + 1.19·19-s + 0.168·20-s + 0.0801·21-s + 0.373·22-s + 1.48·23-s − 0.0572·24-s − 0.887·25-s + 0.529·26-s − 0.319·27-s + 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 0.280T + 3T^{2} \) |
| 5 | \( 1 - 0.751T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 - 0.0598T + 47T^{2} \) |
| 53 | \( 1 - 2.19T + 53T^{2} \) |
| 59 | \( 1 + 0.263T + 59T^{2} \) |
| 61 | \( 1 + 8.78T + 61T^{2} \) |
| 67 | \( 1 + 9.99T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 5.31T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.077866796598695690565905497939, −7.52522321746359976640426490354, −6.79150609906103544669993351274, −5.82013668304628714046311302391, −5.23825796662481078475343542594, −4.35182641305792310324012432423, −2.94191610286367765030346935293, −2.54542392314302492737332287575, −1.36929041066346356440945138518, 0,
1.36929041066346356440945138518, 2.54542392314302492737332287575, 2.94191610286367765030346935293, 4.35182641305792310324012432423, 5.23825796662481078475343542594, 5.82013668304628714046311302391, 6.79150609906103544669993351274, 7.52522321746359976640426490354, 8.077866796598695690565905497939
